Optimal. Leaf size=12 \[ \cos (x)+2 \log (1-\cos (x)) \]
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Rubi [A] time = 0.0362866, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \cos (x)+2 \log (1-\cos (x)) \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{(1-\cos (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1-x}{1+x} \, dx,x,-\cos (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-1+\frac{2}{1+x}\right ) \, dx,x,-\cos (x)\right )\\ &=\cos (x)+2 \log (1-\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0161087, size = 13, normalized size = 1.08 \[ \cos (x)+4 \log \left (\sin \left (\frac{x}{2}\right )\right )-1 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 11, normalized size = 0.9 \begin{align*} \cos \left ( x \right ) +2\,\ln \left ( -1+\cos \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.09661, size = 14, normalized size = 1.17 \begin{align*} \cos \left (x\right ) + 2 \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.593, size = 47, normalized size = 3.92 \begin{align*} \cos \left (x\right ) + 2 \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.645438, size = 61, normalized size = 5.08 \begin{align*} \frac{2 \log{\left (\cos{\left (x \right )} - 1 \right )} \cos{\left (x \right )}}{\cos{\left (x \right )} - 1} - \frac{2 \log{\left (\cos{\left (x \right )} - 1 \right )}}{\cos{\left (x \right )} - 1} + \frac{\sin ^{2}{\left (x \right )}}{\cos{\left (x \right )} - 1} + \frac{2 \cos ^{2}{\left (x \right )}}{\cos{\left (x \right )} - 1} - \frac{2 \cos{\left (x \right )}}{\cos{\left (x \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18609, size = 16, normalized size = 1.33 \begin{align*} \cos \left (x\right ) + 2 \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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